The Witten genus has famously been lifted to the [string orientation of tmf](http://ncatlab.org/nlab/show/string%20orientation%20of%20tmf) ("topological Witten genus"). For the Ochanine genus, I am aware of a lift to a "[spin orientation of Tate K-theory](http://ncatlab.org/nlab/show/spin%20orientation%20of%20Tate%20K-theory)", namely, to a map of ring spectra $M \mathrm{Spin}\to KO[ [q] ]$: lemma 5.4, 5.8 in 

* Matthias Kreck, Stefan Stolz, _$HP^2$-bundles and elliptic homology_, Acta Math, 171 (1993) 231-261

At least naively, one would hope to see a further refinement of the Ochanine genus to an orientation of something like $tmf_0(2)$. Is there any reason for why this should not work? Or has it just not been considered?